Dynamic Viscoelasticity of Actin Networks
Cross-Linked with Wild-Type and Mutant a-Actinin-4
by
Sabine M. Volkmer Ward
Diplom Physics, Friedrich-Schiller-Universitit Jena, 2005
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2008
@ Sabine M. Volkmer Ward, MMVIII. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute
publicly paper and electronic copies of this thesis document in whole or in
part.
Author ........... •0
Department of Physics
Feb 1, 2008
Certified by
/ C~L'Y
Professor
9hysics
and of Applied
David A. Weitz
Physics, Harvard University
Thesis Supervisor
Certified by
G/
1
Alexander van Oudenaarden
Associate Professor of Physics
/Thesis Supervisor
Accepted by.......
/ Tho s J. Greytak
Associate Department Head for Education
MASSACHUSETTS INSTIUTE
OF TECHNOLOGY
OCT 09 2008
L F1BRARIES
ARCHIVES
Dynamic Viscoelasticity of Actin Networks Cross-Linked with
Wild-Type and Mutant a-Actinin-4
by
Sabine M. Volkmer Ward
Submitted to the Department of Physics
on Feb 1, 2008, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
The actin cross-linker a-actinin-4 has been found indispensable for the structural and functional integrity of podocytes; deficiency or alteration of this protein due to mutations disturbs the cytoskeleton and results in kidney disease. This thesis presents rheological studies
of in vitro actin networks cross-linked with wild-type and mutant a-actinin-4, which provide
insight into the effect of the cross-linker on the mechanical properties of the networks.
The frequency dependent viscoelasticity of the actin/a-actinin-4 networks is characterized by an elastic plateau at intermediate frequencies, and relaxation towards fluid properties
at low frequencies. Since the elastic plateau is a consequence of cross-linking, its modulus increases with the a-actinin-4 concentration. Networks with wild-type and mutant a-actinin-4
differ significantly in their time scales: The relaxation frequencies of networks with the mutant cross-linker are an order of magnitude lower than that with the wild-type, suggesting a
slower dissociation rate of mutant a-actinin-4 from actin, consistent with a smaller observed
equilibrium dissociation constant. This difference can be attributed to an additional binding
site, which is exposed as a result of the mutation. An increase in the temperature of networks
with mutant a-actinin-4 appears to return the viscoelasticity to that of networks cross-linked
by the wild-type. Moreover, the temperature dependence of the relaxation frequencies follows
the Arrhenius equation for both cross-linkers. These results strongly support the proposition
that the macroscopic relaxation of the networks directly reflects the microscopic dissociation
rates of their constituents.
Thesis Supervisor: David A. Weitz
Title: Professor of Physics and of Applied Physics, Harvard University
Thesis Supervisor: Alexander van Oudenaarden
Title: Associate Professor of Physics
Acknowledgments
I would like to thank my research advisor, David Weitz, and his group at the School of
Engineering and Applied Sciences at Harvard University, for providing a stimulating and
supportive environment in which the work presented in this thesis could thrive. Dave has
provided insight and numerous fruitful suggestions for this project. Moreover, his enthusiasm,
direct criticism, and genuine support foster a great work atmosphere in the group and beyond,
which makes research exciting and enjoyable. I would also like to thank my co-supervisor,
Alexander van Oudenaarden from MIT, for his support.
This research is the result of a collaboration with several colleagues at Harvard Medical
School. I am particularly grateful to Astrid Weins and Martin Pollak, who introduced me
to the medical findings which motivated my rheological studies of wild-type and mutant
a-actinin-4, provided me with these proteins, and finally discussed my results. Further, I
would like to thank Fumihiko Nakamura from Thomas Stossel's group for purifying the actin
I used, and answering all my biochemical questions promptly and most helpfully. I thank Bill
Brieher from Tim Mitchison's lab for providing chicken gizzard a-actinin for comparative
measurements.
My work has benefited from the help and hints of many "Weitzlab" members. Special
thanks go to Karen Kasza and Dan Blair, who performed the precursor experiments for my
project, and gave me much good advice throughout. I am also indebted to the members of
our Biophysics Working Group as well as the frequent occupants of the rheology lab, whose
ideas and problem solving skills have accelerated my progress many times.
During my stay at Harvard, I have had the opportunity to explore a variety of techniques
in the broader context of my research. I am very grateful to Claudia Friedsam for sharing
her expertise in Atomic Force Microscopy with me, and to Hans Wyss for showing me how to
study macroscopic forces with a beautifully simple microaspiration setup. Further, I would
like to acknowledge the kind help of Martin Vogel, Richard Schalek, and Jiandong Deng,
who provided assistance with the Confocal Microscope and AFM at the Harvard Center for
Nanoscale Systems.
Finally, I would like to thank my husband, David Ward, who is an inexhaustible source
of intellectual stimulation and adventure, and our families in Germany and Texas, visiting
whom always reminds me that "there is no place like home".
Contents
1
13
Introduction
. . . . . . . . . . . . . . .... .
1.1
R heology . . . . . . . . . . . . . . . . . . .
1.2
Actin networks
1.3
Mutant a-actinin-4 and its implications for kidney function ..........
. . . . . . . . . . . . ..
. . . . . . . . . ..
. . . . . . .. .
18
23
27
2 Materials and Methods
. . . . . . . . . . . . . . . . . . .
13
2.1
M aterials
. . . . . . . . . . . . . .. .. .
2.2
Network formation ................................
27
2.3
Network visualization ...............................
28
2.4
Bulk rheology . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . ... . .
27
28
3 Comparison of the structures and viscoelastic properties of actin networks
cross-linked with wild-type and mutant a-actinin-4
29
3.1
Network structures ................................
29
3.2
Overview over the frequency-dependent viscoelasticity of the networks . . . .
30
3.3
Cross-linker concentration and network strength ........
...
33
3.4
Temperature and network dynamics ...................
....
38
4 Conclusion
. .
47
List of Figures
1-1
Rheology with a cone and plate geometry. (a) The sample is loaded between
the lower plate and a conical tool, which rotates around its axis. (b) Definition
of quantities used in equations 1.8, 1.9, and 1.10. The dimensions are not
drawn to scale; cone angle and gap size are exaggerated for clarity...... .
1-2
17
Elastic modulus G' (solid squares) and viscous modulus G" (open squares) of
entangled actin (1 mg/ml). (a) Moduli as a function of time after initialization
of the polymerization reaction, measured at a frequency of 0.1 Hz and a strain
of 0.005. The network stiffness levels off after about 30 minutes. (b) Frequency
dependent viscoelasticity of the network, measured at a strain of 0.02. The
elastic modulus is depends only weakly on the frequency. . ........
1-3
. . 21
Frequency dependent viscoelasticity of networks of actin (1 mg/ml) with and
without cross-linkers. (a) Pure entangled actin (black squares), courtesy of
Karen Kasza; actin with filamin at a molar ratio R = 0.01 (green triangles),
courtesy of Karen Kasza; actin with scruin at R = 0.02 (red circles), courtesy
of Gijsje Koenderink. (b) Pure entangled actin (black squares); actin with
chicken-gizzard a-actinin at R = 0.01 (blue inverted triangles); actin with
human a-actinin-4 at R = 0.01 (magenta diamonds). All samples in (b) are
prepared with actin from the same batch ...................
.
.
22
3-1
Confocal images of networks formed of actin (1 mg/ml) (a) without crosslinker, (b) with wild-type a-actinin-4 (R = 0.01), and (c) with mutant aactinin-4 (R = 0.01). Actin is fluorescently labeled with Alexa-Flour-Phalloidin
. . 30
(excitation at 488 nm). Image size: 153.6 pm x 153.6 pm. . ........
3-2
Frequency dependence of the elastic modulus G' (solid symbols) and the viscous modulus G" (open symbols) of actin networks with wild-type (black
circles) and mutant (red triangles) a-actinin-4 at a molar ratio of a-actinin-4
to actin of R = 0.01. The networks are characterized by an elastic plateau,
and a relaxation towards fluid properties at lower frequencies. The response
for mutant a-actinin-4 appears shifted towards lower frequencies, and exhibits
a higher ratio G'/G" at the plateau frequency. . ................
3-3
31
Frequency dependence of the elastic modulus G' (solid symbols) and the viscous modulus G" (open symbols) of actin networks with (a) wild-type and (b)
mutant a-actinin-4 at molar ratios of a-actinin-4 to actin of R = 0.005 (red
triangles), R = 0.01 (black circles), and R = 0.02 (blue inverted triangles).
For both types of cross-linker, the elastic plateau modulus increases with concentration. Moreover, the plateau is more pronounced and appears at higher
frequencies for increased concentration. . .................
3-4
.
.
.
34
The frequency dependence of G' (solid symbols) and G" (open symbols) of
actin networks with wild-type (black circles) and mutant (red triangles) aactinin-4 of different concentrations can be scaled onto one curve. For example, samples with wild-type at high concentration (R = 0.04) and mutant at
medium concentration (R = 0.01), as shown in (a), overlap if the mutant is
shifted by a factor of 10 in frequency and by a factor of 8.7 in the moduli
(b). Similarly, samples with wild-type at medium concentration (R = 0.01)
and mutant at low concentration (R = 0.0025), shown in (c), overlap if the
mutant is shifted by a factor of 14 in frequency and by a factor of 3.9 in the
moduli (d).
....................................
35
3-5
Characteristic features of the frequency-dependent viscoelasticity as a function of the relative concentration R of wild-type (black circles) and mutant
(red triangles) a-actinin-4. (a) The plateau modulus G',ateau has comparable
values for wild-type and mutant, and increases with cross-linker concentration.
Above R = 0.005, the plateau modulus follows a power law for both crosslinker types, as indicated by the solid lines for wild-type (black) and mutant
(red) and their respective slopes (lower right). (b) The ratio of the plateau
modulus and the corresponding viscous modulus, (G'/G")plateau, grows weakly
with increasing cross-linker concentration, and is on average higher for the mutant than for the wild-type at the same concentration. (c) The frequency at
which the elastic plateau occurs (minimum in G") increases weakly with crosslinker concentration, and is systematically smaller for the mutant. (d) The
relaxation frequency of the network (maximum in G") appears to be independent of the cross-linker concentration, and is about an order of magnitude
smaller for the mutant.
3-6
..
...................
.........
37
Frequency dependence of the elastic modulus G' (solid symbols) and the viscous modulus G" (open symbols) of actin networks with (a) wild-type and
(b) mutant a-actinin-4 (molar ratio of a-actinin-4 to actin R = 0.01) at 37 0 C
(red triangles), 25 C (black circles), and 15'C (blue inverted triangles). Similarly to a high cross-linker concentration, a lower temperature results in a
more pronounced elastic plateau. In addition, a temperature decrease shifts
the network relaxation towards lower frequencies.
. ...............
39
3-7 An increase in temperature can return the viscoelasticity of networks with
mutant a-actinin-4 (red triangles) to that of networks with wild-type crosslinker (black circles), as illustrated by the frequency dependence of G' (solid
symbols) and G" (open symbols) for (a) R = 0.01, wild-type at 25 C, mutant
at 37 0 C, and (b) R = 0.02, wild-type at 15 C, mutant at 32.5 oC. In contrast
to the scaling behavior for mutant and wild-type at different concentrations,
a shift in frequency or moduli is not necessary. . ..............
3-8
. .
41
Temperature dependence of the network relaxation frequency. The logarithm
of the frequency is plotted as a function of the inverse thermal energy. For both
wild-type (black circles) and mutant (red triangles) cross-linker, the relaxation
frequency follows the Arrhenius equation. The slopes, which correspond to
an activation energy barrier in the Arrhenius equation, are comparable. Relaxation frequencies for the mutant are consistently lower by about an order
of magnitude. ...................
...............
..
43
Chapter 1
Introduction
This thesis presents rheological studies of in vitro networks formed of actin and a-actinin-4,
two physiologically very important proteins. Rheometry, a technique commonly used to
measure the deformation and flow properties of soft materials for example in the food,
cosmetic, and oil industry, has been discovered over the past decades as a means to investigate
the characteristics of biological materials as well [11]. In particular, it has helped to elucidate
the mechanical properties of the cytoskeleton and its constituents: actin, microtubules, and
intermediate filaments. Recent medical interest in the actin cross-linker a-actinin-4, whose
lack or mutation has been found to cause kidney disease, has motivated the research presented
here [25]. To put the results of these experiments in context, the following sections will
provide background information on rheology, actin networks, and a-actinin-4 and its critical
role for the structural and functional integrity of kidney cells.
1.1
Rheology
Rheology (from Greek pew = to flow) is the study of the deformation and flow of soft condensed matter in response to external forces. The term applies generally to both extensional
rheology, in which the force per unit area, or stress, is normal to the face of the material,
and shear rheology, in which the stress is tangential to the surface. In the following, we will
only deal with shear rheology. We denote the shear stress with a, and the resulting shear
strain, defined as the tangent of the shear angle, with y, as is conventional.
Classically, condensed matter is classified into liquids and solids. When liquids are subjected to shear stress, they flow, i.e. they deform irreversibly, dissipating the deformation
energy as heat. In the ideal case of a Newtonian liquid, the rate of deformation, or strain
rate dy/dt, is proportional to the applied stress:
a=
dy
-d
(1.1)
The constant of proportionality, q, is the dynamic viscosity of the material, and can be
thought of as a measure of liquid friction. Water is a thin liquid with a viscosity of about
1 mPa.s, whereas olive oil is thicker and has a viscosity of 81 mPa-s. Solids under stress
deform reversibly until an equilibrium between the external forces and the materials restoring
forces is reached. The deformation energy is stored and, for an ideal elastic solid, completely
recovered once the external stress is released. Here, the deformation, not its time derivative,
is proportional to the applied stress:
a= G -y
(1.2)
The constant of proportionality is the shear modulus G. As an example, commercial Jello
has a shear modulus of -100 Pa.
Models that capture the idealized behaviors of a Newtonian liquid and an elastic solid
are a dash pot and a Hookian spring, respectively. Many materials we encounter in everyday
life, however, do not completely fall in either of these categories. Instead, they exhibit both
fluid-like and solid-like behaviors-they are viscoelastic [14].
Illustrative examples of such
complex fluids are mayonnaise and peanut butter, shampoo and toothpaste, drilling mud,
blood, and liquid crystals. Viscoelastic materials can be modeled by a combination of springs
and dash pots. Their deformation, or strain (-y), generally lags behind the applied stress.
Consequently, the ratio between stress and strain can be described by a complex number,
the dynamic modulus G*:
= G* = G' + i G"
(1.3)
The real part of G* is the elastic or storage modulus G'; the imaginary part is the viscous or
loss modulus G". For an ideal elastic solid, stress and strain are in phase, and the dynamic
modulus reduces to G'. For a Newtonian liquid, stress and strain are exactly 900 out of
phase, such that the dynamic modulus reduces to i G". The relation between G* and the
viscosity 7 and shear modulus G introduced in equations 1.1 and 1.2 is best illustrated for
a sinusoidally oscillating strain:
7 = y
.exp(i wt)
(1.4)
d-y
dt = G' -
e- xp(i w
(1.5)
Gt)
w
w dt
It follows that G' = G and G" = r7 w. As we see, the viscous modulus G" is proportional
to the frequency of the imposed mechanical perturbations for an ideal liquid of frequency
independent viscosity.
In general, however, both the viscosity and the elasticity of the
network depend on the rate of deformation, as they reflect the properties and typical time
scales of structural rearrangements in the material. Taking the imaginary part of equations
1.4 and 1.5, stress and strain are therefore related by:
a(t) = 'Yo [G'(w) sin(w t) + G"(w) cos(w t)]
(1.7)
Matters are further complicated by the fact that the linear viscoelastic response described by
equation 1.7 only holds for small perturbations which leave the network properties unaltered.
At high stresses, the dynamic modulus can itself be stress dependent. If it increases with
stress, leading to a superlinear increase of the applied stress amplitude with the strain amplitude, the material is called stress-stiffening. If, on the other hand, the modulus decreases
at high stresses, resulting in a sublinear increase of stress with strain, the material yields
and is described as stress-weakening. In the non-linear regime, a sinusoidally varying stress
does not result in a sinusoidal strain any more, and vice versa.
The dynamic modulus of a soft material can be measured with a conventional rheometer,
which subjects the sample to a shear stress, and determines the resulting shear strain. Alternatively, the instrument can impose a specific strain or constant shear rate, and measure
the resulting stress. These two modes of operation are referred to as stress- and straincontrolled, respectively. Rheometers can control shear stress over a large range, from 0.01 Pa
to 10 kPa, which, for soft materials, extends well into the non-linear regime. The sample is
typically loaded into a cylinder-symmetric geometry, such as the space between to concentric
cylinders, or between two horizontal plates. For the experiments described later on in this
thesis, a cone and plate geometry, as illustrated in Fig. 1-1(a), was used. Through rotation
of the conical tool around its axis, a torsional shear stress is applied to the sample. This
specific geometry bears the advantage of admitting uniform shearing: If a tool with cone
angle 6 rotates by an angle q, a cylindrical ring of the sample with radius r and height h(r)
experiences a shear strain
=
r
h(r)
=tan 6'
(1.8)
_
which is independent of its radius (Fig. 2b). The shear stress is likewise uniform across the
sample, and results in a differential force
dF(r) = r - 27 r dr
and a torque
R
o
7- =
[r dF(r) =
27r
-
(1.9)
,
(1.10)
where R is the radius of the tool. From the exerted torque T and the resulting angular displacement ¢ of the tool, the instrument computes stress and strain according to equations
1.8 and 1.10.
Rheometry can be static or oscillatory. An example of the former is a creep experiment,
in which a constant shear stress is applied, and the transient strain or strain rate is monitored
(a)
.:-i
1-4
(b)
R
6.......
Ih(r)
:4
_
Figure 1-1: Rheology with a cone and plate geometry. (a) The sample is loaded between
the lower plate and a conical tool, which rotates around its axis. (b) Definition of quantities
used in equations 1.8, 1.9, and 1.10. The dimensions are not drawn to scale; cone angle and
gap size are exaggerated for clarity.
over time. After removal of the applied stress, the relaxation of the material can be observed
similarly. For liquid-like materials, the shear rate is often held constant, and the time
dependent stress is measured until a steady stress is reached. These types of experiments
provide information about the rheological properties on long time scales. Here, we present
the results of small-amplitude oscillatory rheology, and focus on the frequency dependence
of the linear viscoelasticity, as described by equation 1.7, for frequencies between 0.01 and
10 Hz.
The limitations of conventional rheology, in particular when applied to biological materials, lie in the need for sample volumes that are large compared with the quantities
in which proteins are typically purified, and in the effects of sample drying and protein denaturation, which limit the duration of measurements to several hours. Further, bulk rheology provides only sample-averaged data, whose meaningfulness depends on the homogeneity
of the sample.
A complementary approach to studying the properties of soft materials
is microrheology, in which micron-sized colloidal spheres, imbedded in the material and
optically tracked, probe the local thermal strain fluctuations of the network. Microrheology
bears the advantages of higher accessible frequencies (up to 105 Hz) and smaller sample
volumes. However, it is practically limited to samples with elastic moduli of no more than
10 Pa, and the interpretation of the thermal motion of the tracer colloids can be complicated
through heterogeneities in the network microstructure. The actin/a-actinin networks studied
in this thesis span a range of stiffnesses between one and hundreds of Pascals, and exhibit very
interesting features at shear frequencies around 1 Hz. Bulk rheology therefore constitutes an
adequate method for the investigation of their viscoelastic properties.
1.2
Actin networks
Cells owe their shape, spatial organization, motility, adhesion, and mechanical stability to
an intricate network of interconnected filamentous proteins, called the cytoskeleton [1]. The
major building blocks of the cytoskeleton are three types of polymers, each assembling into
characteristic structures and carrying typical functions: Actin filaments form bundles and
networks close to the cell membrane, which determine the cell shape and enable cell locomotion with the help of surface protrusions. In muscle cells, they arrange in parallel with
myosin to build long, contractile fibers. Microtubules often occur in a star-like configuration
and are responsible for the positioning and intracellular transport of cell organelles. Intermediate filaments, which can be grouped into several families located in different cell types,
provide mechanical stability and resistance to shear stress and pressure.
In an attempt to gain further insight into the mechanical properties of cells and the cytoskeleton, researchers have studied two types of systems: the cells themselves, and model
networks formed of individual cytoskeletal components in vitro [11]. Much research in recent years has focused on reconstituted networks of actin-one of the most abundant proteins in eukaryotic cells. Filamentous actin, or F-actin, is a polymer composed of 42 kDa
globular monomeric subunits called G-actin. At physiological salt concentrations, G-actin
self-assembles into double-stranded, helical, semiflexible micro-filaments with a 72 nm pitch
and a filament diameter of approximately 7 nm [17]. Actin filaments are polarized, dynamic
structures. In equilibrium, they assemble at one end and disassemble at the other end,
reaching lengths of typically 20 pm. The persistence length of actin filaments, defined as
the length over which correlations in the tangent vector to the filament contour are lost,
is comparable (-17 pm) [4]. Therefore, actin filaments behave neither as rigid rods nor as
completely flexible polymers; they are semiflexible, and their thermal bending fluctuations
contribute to the network properties, in particular at short time scales.
Purified actin polymerized in vitro at concentrations of -1 mg/ml forms finely meshed
entangled networks, as shown by confocal microscopy of fluorescently labeled actin (Fig. 31(a)). Rheologically, these networks behave like weak viscoelastic solids. Upon initialization
of the polymerization reaction, the network stiffness increases rapidly over a few up to tens of
minutes, reflecting the growing fraction of polymerized actin, and then starts leveling off (Fig.
1-2(a)). The elastic modulus of the final network is only weakly frequency dependent over a
large range (Fig. 1-2(b)). It typically assumes values on the order of 0.1 Pa to 1 Pa. While
two orders of magnitude higher moduli have been reported for entangled actin, consensus has
been reached that this discrepancy is largely a consequence of protein aging under unsuitable
storage conditions [26], and partly a result of traces of actin-binding proteins that may
exist as impurities in the solutions, as well as of repeated mechanical perturbations during
measurements [9]. The data presented in Fig. 1-2 are taken on actin samples prepared under
the same conditions as those studied later on in this thesis, up to the deliberate addition of
cross-linkers to the latter, and serve for comparisons of pure and cross-linked networks.
The actin cytoskeleton in cells is orders of magnitude stiffer than pure entangled actin
networks in vitro, suggesting that cross-linking is essential in inducing physiologic network
properties. In vivo, a myriad of cross-linker, motor, and other actin binding proteins regulate
the dynamic actin polymerization and the formation of higher order structures [1, 2]. This
complexity renders the detailed understanding of the actin cytoskeleton extremely difficult.
Researchers therefore typically limit their in vitro studies of reconstituted actin networks to
the investigation of one or two regulatory proteins and their effect on the network properties.
One such accessory protein is scruin, a rod-like, rigid 102 kDa molecule, which cross-links
and bundles actin irreversibly into isotropic, homogeneous three-dimensional networks. At
molar ratios of scruin to actin monomers from 0 and 1, scruin increases the elastic modulus of
these networks over three orders of magnitude [7, 19]. Since scruin itself does not contribute
to the network compliance, actin-scruin networks serve as a model system for studying
the network elasticity that result from the properties of individual actin filaments. Two
prominent and physiologically relevant compliant cytoskeletal proteins are filamin and aactinin. Filamin is a hinged dimer of 560 kDa with two actin binding domains, which crosslinks actin microfilaments into orthogonal networks. At physiological molar ratios to actin
of 0.01 to 0.02, filamin does not affect the linear elasticity of the network much; however,
it results in drastic strain stiffening, that is, a remarkable increase in the differential elastic
modulus, K = R(dou/dy), at strains above 10% [5, 6]. Fig. 1-3(a) shows the linear viscoelastic
response of pure entangled actin, and networks cross-linked with scruin (red circles) and
filamin (green triangles), illustrating the different effects of the two cross-linkers on the
linear network properties.
_
II
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(a)
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Figure 1-2: Elastic modulus G' (solid squares) and viscous modulus G" (open squares)
of entangled actin (1mg/ml). (a) Moduli as a function of time after initialization of the
polymerization reaction, measured at a frequency of 0.1 Hz and a strain of 0.005. The
network stiffness levels off after about 30 minutes. (b) Frequency dependent viscoelasticity
of the network, measured at a strain of 0.02. The elastic modulus is depends only weakly on
the frequency.
(a)
CU 10
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Figure 1-3: Frequency dependent viscoelasticity of networks of actin (1
mg/ml) with and
without cross-linkers. (a) Pure entangled actin (black squares), courtesy of Karen Kasza;
actin with filamin at a molar ratio R = 0.01 (green triangles), courtesy of Karen
Kasza; actin
with scruin at R = 0.02 (red circles), courtesy of Gijsje Koenderink. (b) Pure
entangled actin
(black squares); actin with chicken-gizzard a-actinin at R = 0.01 (blue inverted triangles);
actin with human a-actinin-4 at R = 0.01 (magenta diamonds). All samples in (b) are
prepared with actin from the same batch.
The antiparallel homodimer a-actinin falls somewhere in between scruin and filamin regarding its molecular size and structure. It weighs 200 kDa and measures 40 nm in length, is
a rigid rod like scruin, but binds reversibly to actin at its two diametric binding sites like filamin. Depending on its concentration, a-actinin can cross-link or bundle actin filaments [21],
or form networks of bundles with mesh sizes on the order of the persistence length of actin
[18]. Compared with other cross-linkers, it tends to cause greater network inhomogeneities.
The various isoforms of the cross-linker, which can be found in smooth muscle, skeletal
muscle, and non-muscle cells, result in networks of widely varying viscoelastic properties
[21, 27, 20]. Chicken gizzard a-actinin, for example, increases the stiffness of entangled actin
only slightly, whereas a-actinin-4, one of the four isoforms found in mammals, drastically
alters the frequency dependent viscoelasticity, as shown in Fig. 1-3(b). The actin binding
dynamics of a-actinin in its effect on the network properties has been studied by several
authors with chicken and turkey gizzard as well as Acanthamoeba a-actinin [21, 27, 20, 22].
Recently, human a-actinin-4 has gained the attention of medical scientists in the context of
research on kidney disease.
1.3
Mutant a-actinin-4 and its implications for kidney
function
Deficiency or mutations of the gene that encodes a-actinin-4 can cause Focal and Segmental
Glomerulosclerosis (FSGS), a pattern of renal injury, which is characterized by increased
urinary protein excretion and slowly progressive kidney dysfunction [10, 12, 15, 24]. An
estimated 4% of familial FSGS in affected humans are accounted for by these mutations [24].
Tissue damage is often a consequence of the collapse of glomeruli-renal capillary tufts which
serve blood filtration through a three-layered system comprised of endothelium, glomerular
basal membrane, and epithelium. Glomeruli are subject to high blood pressures that aid in
the process of ultrafiltration; moreover, glomerular hypertension has been established as a
source of renal failure. Research on kidney disease has therefore included investigations of
the mechanical properties of glomeruli and the filtration barrier [13, 3].
In a-actinin-4 mediated FSGS, the epithelial podocytes are believed to be the initial
sites of injury [12].
Podocytes form an interdigitating network of foot processes, which
is a major constituent in the filtration barrier, and are exposed to significant mechanical
stress due to changes in glomerular capillary pressure. In these cells, a-actinin-4, which is
believed to regulate their highly organized actin cytoskeleton, is highly expressed. Since the
unique structure of podocytes renders them particularly susceptible to changes in the actin
cytoskeleton, a-actinin-4 mutations can have drastic consequences for the kidney. To date,
five disease-causing point mutations in the actin binding domain of a-actinin-4 have been
identified [24]. All of these mutations increase the protein's actin binding affinity, and lead
to abnormal actin aggregation and foot process effacement.
One particular point mutation, which encodes a K255E amino acid exchange in the
a-actinin-4 protein and is known to cause FSGS and kidney failure in humans, has been
extensively investigated by Weins et al. [25]. A mouse model homozygous for the mutation
exhibits large actin/a-actinin-4 aggregates in the podocytes, and subsequently develops a
form of collapsing glomerulopathy.
In standard cosedimentation assays of actin and a-
actinin-4 , the equilibrium dissociation constant for the mutant has been determined to be
Kd,mut
=
0.046 pM, almost six times lower than that of the wild-type, Kd,wt = 0.267 pM.
Further, the mutant saturates actin at one dimer per two actin monomers, whereas the wildtype saturates at one dimer per four actin monomers. Weins et al. attribute these differences
to a conformational change in the protein. Each a-actinin monomer is composed of an Nterminal F-actin binding domain, four spectrin repeats, and a C-terminal Ca 2 + binding
domain. In wild-type a-actinin-4, the actin binding domain contains two actin binding sites,
and its actin affinity can be regulated by Ca 2+ binding to the neighboring N-terminal domain
of the second a-actinin-4 monomer. An additional actin binding site is partially buried in the
"closed" conformation of the wild-type. The K255E mutation is believed to lead to a higher
propensity of the open conformation, which exposes the otherwise buried actin binding site.
The mutant has also been shown to be insensitive to regulation by Ca 2+ . This explanation is
corroborated by the fact that inactivation of the additional binding site returns the affinity
of the mutant to that of the wild-type, while leaving the affinity of the wild-type unaltered.
Weins et al. propose that the different stoichiometric ratios of a-actinin-4 to actin can
be ascribed to greater geometric flexibility of the mutant, which allows it to access every
binding site along an actin filament. This flexibility also serves to explain their observation
that actin polymerized in the presence of mutant a-actinin-4 forms networks of thin actin
bundles, whereas the wild-type tends to connect actin into thicker, parallel bundles. Using
miniature falling ball viscometry, Weins et al. have found that the mutant results in an
abrupt viscosity increase, signifying the gel point of the network, at a much lower cross-linker
concentration than the wild-type [25]. Further comparative investigations of the viscosity
and elasticity of networks with wild-type and mutant a-actinin-4 can provide more insight
into the disease mediating characteristics of the mutant, especially in view of the pivotal
role of the actin cytoskeleton for podocytes; however, systematic studies of actin networks
cross-linked specifically with a-actinin-4 have not been performed previously.
This thesis provides a detailed analysis, based on rheological data, of the viscoelastic
properties of in vitro networks formed from actin with wild-type and K255E-mutant aactinin-4. The difference between the two cross-linkers is most apparent in measurements of
the elastic and viscous moduli as a function of the frequency of the applied oscillatory shear
stress (section 3.2). The effect of different binding affinities on the macroscopic network
dynamics is contrasted with the effect of varied cross-linker concentration (section 3.3) and
temperature (section 3.4).
Chapter 2
Materials and Methods
2.1
Materials
Actin was provided by Fumihiko Nakamura and Thomas Stossel (Brigham and Women's Hospital, Harvard Medical School), who prepared it from rabbit skeletal muscle in a polymerizationdepolymerization cycle, and subsequently further purified it by gel filtration. G-actin was
stored in G-buffer (2 mM Tris-HC1, pH 7.4, 0.5 mM ATP, 0.5 mM 2-Mercaptoethanol, 0.2 mM
CaC12) at -80 oC. Wild-type and mutant a-actinin-4 were provided by Astrid Weins and Martin Pollak (Brigham and Women's Hospital, Harvard Medical School). Baculovirus-expressed
recombinant wild-type and K255E mutant a-actinin-4 was generated from human cDNA sequences by ProteinOne (College Park, MD, USA) [25] and stored in a-actinin buffer (20 mM
Tris-HC1, pH 7.9, 20% Glycerol, 100 mM NaC1, 1 mM DTT, 0.2 mM EDTA) at -80 0 C.
2.2
Network formation
Networks of various cross-linker concentrations were formed by mixing 23.8 jM (1 mg/ml) Gactin solution with a-actinin-4 solutions of concentrations ranging from 0.03 pM to 1.9 pM,
corresponding to molar ratios of a-actinin-4 to actin, R, between R = 0.00125 and R =
0.08. Polymerization was initiated by addition of 5x polymerization buffer (50 mM Tris-HC1,
pH 7.4, 500mM KC1, 5mM MgC1, 2.5mM EGTA, 2.5mM ATP). The samples were then
immediately transferred to the rheometer or confocal microscope, respectively.
2.3
Network visualization
Actin was fluorescently stained with 2 pM Alexa Fluor 488 phalloidin (Invitrogen). Samples
of 10 pl1 in volume were confined between two glass cover slips spaced at '-1 mm, and examined
under a Zeiss LSM 510-Meta confocal microscope with a 63x objective. Fluorescence was
excited with an Argon laser at 488 nm, and the signal was filtered by a 505 nm long pass
filter.
2.4
Bulk rheology
Oscillatory measurements of the elastic modulus G' and the viscous modulus G" were performed with a controlled stress AR-G2 rheometer (TA Instruments). The sample was loaded
into a cone-plate geometry with a diameter of 20 mm, a cone angle of 20, and a gap size of
50 pm; the sample volume was 70 pl. A solvent trap was used to prevent drying. Immediately
following sample transfer to the rheometer, polymerization was monitored for a period of one
to two hours through point measurements of the viscoelastic moduli in five minute intervals
at a strain of 0.5% and an oscillation frequency of 0.1 Hz. The frequency dependence of the
linear viscoelastic moduli G' and G" was determined between 0.01 Hz and 10 Hz. To ensure
that data was taken in the linear regime, where the network response is proportional to
the amplitude of the imposed oscillatory stress, the instrument was operated in its straincontrolled mode at strains between 0.5% and 2%, depending on the strength of the network.
For each measurement of the frequency dependence, the temperature was held constant, if
not stated otherwise, at 25 'C. To investigate the temperature dependence of the network
dynamics, the temperatures for different measurements at the same sample were varied between 5 oC and 40 C by means of the Peltier temperature control of the lower rheometer
plate.
Chapter 3
Comparison of the structures and
viscoelastic properties of actin
networks cross-linked with wild-type
and mutant a-actinin-4
3.1
Network structures
Actin polymerized in vitro forms finely meshed, homogeneous, isotropic three-dimensional
networks (Fig. 3-1(a)). The addition of cross-linking proteins often results in networks of
larger mesh size, which consist of actin bundles comprising many individual filaments. We
confirmed by confocal fluorescence microscopy that this general trend also holds for networks
of actin with wild-type and mutant a-actinin-4 , prepared in the same way as the samples
used for our rheological measurements. At a concentration of one a-actinin-4 molecule per
hundred actin monomers, R = 0.01, the wild-type cross-linker (Fig. 3-1(b)) appears to form
a coarser network than the mutant (Fig. 3-1(c)), consistent with the larger bundle diameter
reported by Weins et al. for actin cross-linked with wild-type a-actinin-4. However, we
find the cross-linked networks interspersed with large actin agglomerates (Figs. 3-1(b) and
of actin (1 mg/ml) (a) without cross-linker,
formed
networks
of
images
= 0.01).
Figure 3-1: Confocal
(c) with mutant a-actinin-4 (R
and
0.01),
=
(R
(b) with wild-type a-actinin-4
(excitation at 488nm). Image
Alexa-Flour-Phalloidin
with
labeled
Actin is fluorescently
size: 153.6 pm x 153.6 pm.
increase with higher
sample to sample, and tend to
from
vary
number
and
size
(c)), whose
vary in mesh size
homogeneous network regions
the
Moreover,
cross-linker concentration.
effects of wild-type
a quantitative comparison of the
rendering
thus
sample,
one
even within
unfeasible.
and mutant on the network structure
3.2
viscoelasOverview over the frequency-dependent
ticity of the networks
inhomoof the network structures and
variations
sample-to-sample
large
In spite of the
networks exhibit robust
of micrometers, the cross-linked
tens
of
scales
length
at
geneities
At a moaveraging over larger dimensions.
sample
of
consequence
rheological properties-a
stiffness by about an order of
network
the
increases
lar ratio of R = 0.01, cross-linking
of the applied
vary greatly with the frequency
moduli
viscous
and
elastic
magnitude. The
through bulk
frequencies which we can access
intermediate
the
At
stress.
shear
oscillatory
a nearly frequencyviscoelasticity is characterized by
linear
the
measurements,
rheological
minimum of the viscous
modulus G', and a corresponding
independent plateau in the elastic
_ _ ______
_ _ ______
_
_
U10
_ _____
0. .000
00 A AAA j
000000000
O ,
A.
AA.
AAAAA
)V
__I
'
0.1
1
10
f [Hz]
Figure 3-2: Frequency dependence of the elastic modulus G' (solid symbols) and the viscous
modulus G" (open symbols) of actin networks with wild-type (black circles) and mutant (red
triangles) a-actinin-4 at a molar ratio of a-actinin-4 to actin of R = 0.01. The networks
are characterized by an elastic plateau, and a relaxation towards fluid properties at lower
frequencies. The response for mutant a-actinin-4 appears shifted towards lower frequencies,
and exhibits a higher ratio G'/G" at the plateau frequency.
modulus G" (3-2). A direct comparison of the effect of wild-type (black circles) and mutant
(red triangles) cross-linker reveals that the latter results in a more distinct elastic plateau.
Moreover, the characteristic frequency of the plateau, fplateau, which we define as the frequency at which G" assumes its local minimum, is smaller for the mutant than for the
wild-type.
In the high frequency limit, we expect the network viscoelasticity to be dominated by the
thermal bending fluctuations of individual filaments, leading to an increase of both moduli
with frequency with a power of 3/4 [16, 8]. We cannot measure at frequencies high enough
to confirm this dependence, but we observe for several samples the onset of an ascent of G'
and G" with an increasing slope of up to 0.6 in a log-log plot.
At low frequencies, the networks carry the signature of a structural relaxation process:
The viscous modulus G" approaches the elastic modulus with decreasing frequency, passing
through a maximum value and then decreasing along with G'. We expect G" to eventually
exceed G', indicating a relaxation of the network behavior to fluid properties. While for most
data sets, the range of our measurements does not extend to low enough frequencies to verify
this cross-over, we do observe it in a few cases. For comparisons of relaxation frequencies
across samples, we take frelax to be the well-defined frequency at which G" reaches a local
maximum (3-2).
In comparing networks with wild-type and mutant a-actinin-4, we find
that frelax is more than an order of magnitude lower for the mutant. Taken together with
the smaller equilibrium dissociation constant of mutant a-actinin-4 [25], this observation
suggests that the relaxation of the macroscopic network might be related to the microscopic
unbinding of the cross-linker. When an a-actinin-4 dimer dissociates from actin at either
binding site, the cross-link breaks, and filaments can locally slide with respect to each other.
The cross-linker may bind again to actin at both ends, thereby reforming a link in the
network. If this happens before other cross-linkers in the vicinity of the broken link detach,
a macroscopic relaxation of the network is not possible. However, our data suggests that
sufficiently many cross-linkers unbind simultaneously to allow the network to structurally
relax, and to become fluid-like at low frequencies.
3.3
Cross-linker concentration and network strength
We extend our comparison of wild-type and mutant a-actinin-4 by investigating the effect
of the cross-linker concentration on the frequency dependent viscoelasticity of the networks.
With increasing concentration of wild-type or mutant cross-linker, G' and G" rise noticeably
over the entire frequency range of our measurements. While the elastic modulus is shifted
upwards without a change in its frequency dependence, an increase in concentration does
alter the frequency dependence of the viscous modulus. At the plateau, the increase in G'
with concentration tends to be greater than that in G". Thus, increasing the a-actinin-4
concentration enhances the plateau, pushing the network properties further towards that of
a solid, as illustrated with representative network responses for relative a-actinin-4 concentrations of R = 0.005 (red triangles), 0.01 (black circles), and 0.02 (blue inverted triangles)
in Fig. 3-3(a) for the wild-type and in Fig. 3-3(b) for the mutant cross-linker. Interestingly,
an increased amount of wild-type ac-actinin-4 effects the shape of the viscoelasticity in the
vicinity of the plateau qualitatively similarly to a replacement of wild-type by mutant crosslinker. In comparing the curves in Figs. 3-3(a) and 3-3(b), we see that the behavior for
the wild-type cross-linker of the highest concentration shown resembles that of the mutant
with the lowest concentration, albeit shifted along both axes. Based on this observation, we
investigate whether the altered properties of the mutant can partly be compensated for by
a decreased concentration. We find that this is indeed the case: The frequency dependent
viscoelasticity of a network with wild-type a-actinin-4 at a high concentration of R = 0.04
(black circles) and a network with mutant a-actinin-4 at a four times lower concentration
(R = 0.01, red triangles), shown in 3-4(a), overlap remarkably well if the mutant curve is
shifted towards higher frequencies and higher viscoelastic moduli, as illustrated in Fig. 34(b). Similarly, the curves for samples with a medium amount of wild-type (R = 0.01) and
a low amount of mutant (R = 0.0025), which are shown in Fig. 3-4(c), can be scaled nearly
perfectly (Fig. 3-4(d)). This intriguing scaling behavior might have profound implications
for the understanding of a-actinin-4 in its effect on the network properties.
In order to test the statistical significance of our qualitative observations, we extract
100
(D
I
10
I
100
C1
CL
10
(Q
03
0.01
0.1
1
10
Figure 3-3: Frequency dependence of the elastic modulus G' (solid symbols) and the viscous
modulus G" (open symbols) of actin networks with (a) wild-type and (b) mutant a-actinin4 at molar ratios of a-actinin-4 to actin of R = 0.005 (red triangles), R = 0.01 (black
circles), and R = 0.02 (blue inverted triangles). For both types of cross-linker, the elastic
plateau modulus increases with concentration. Moreover, the plateau is more pronounced
and appears at higher frequencies for increased concentration.
100
w
10
100
10
1
0.01
0.1
1
f[Hz]
10
100
0.01
0.1
1
10
100
f[Hz]
Figure 3-4: The frequency dependence of G' (solid symbols) and G" (open symbols) of actin
networks with wild-type (black circles) and mutant (red triangles) a-actinin-4 of different
concentrations can be scaled onto one curve. For example, samples with wild-type at high
concentration (R = 0.04) and mutant at medium concentration (R = 0.01), as shown in
(a), overlap if the mutant is shifted by a factor of 10 in frequency and by a factor of 8.7 in
the moduli (b). Similarly, samples with wild-type at medium concentration (R = 0.01) and
mutant at low concentration (R = 0.0025), shown in (c), overlap if the mutant is shifted by
a factor of 14 in frequency and by a factor of 3.9 in the moduli (d).
several characteristic quantities from the frequency dependent viscoelasticity of a larger
set of samples, comprised of 24 networks with wild-type and 18 with mutant a-actinin4 and containing the samples presented in Figs. 3-3 and 3-4, and plot these quantities
as a function of cross-linker concentration.
We confirm that the elastic modulus at the
plateau frequency, G'1ateau, increases with cross-linker concentration for both wild-type (black
circles) and mutant (red triangles), as shown in Fig.
3-5(a).
Above relative a-actinin-
4 concentrations of R = 0.005, the plateau modulus follows a power law of G',iateau R4/ 3
(solid lines). A similar concentration dependence has previously been observed for other
cross-linkers [23, 7].
We infer from Fig. 3-5(a) that the values of G',ateau for wild-type
and mutant a-actinin-4 at the same concentration are comparable. This might at first seem
surprising; however, if we assume a first order binding reaction between actin and a-actinin-4,
we calculate that at an actin monomer concentration of 23.8 AM, almost all cross-linkers are
bound to actin. The higher binding affinity of the mutant therefore results only in a marginal
increase of bound a-actinin-4 from 98.8% to 99.8% of the total cross-linker amount.
While the mutant does not appear to increase the overall network strength, it does render
the network more solid-like, as is evident from the consistently higher values for the ratio
of G' and G" at the plateau, which is plotted in Fig. 3-5(b).
As observed before and
confirmed by the larger data set, this ratio increases slightly with concentration. In addition
to G'/G" at the plateau frequency, the prominence of the elastic plateau is characterized
by the width of the frequency range spanned between frelax and fplteau.
We find that
the plateau frequency increases with cross-linker concentration (Fig. 3-5(c)), whereas the
relaxation frequency remains approximately constant (Fig. 3-5(d)). As a result, the plateau
tends to be the wider, the higher the amount of ac-actinin-4. Furthermore, networks with
the mutant cross-linker exhibit a slightly lower plateau and a significantly lower relaxation
frequency. Consequently, the plateau width is greater for the mutant.
0I
a)P
CL
P_
N)
Z.
0
C4
0
a)
N a)
I
x
a)
1
R
Figure 3-5: Characteristic features of the frequency-dependent viscoelasticity as a function of
the relative concentration R of wild-type (black circles) and mutant (red triangles) a-actinin4. (a) The plateau modulus Gpl,ateau has comparable values for wild-type and mutant, and
increases with cross-linker concentration. Above R = 0.005, the plateau modulus follows a
power law for both cross-linker types, as indicated by the solid lines for wild-type (black) and
mutant (red) and their respective slopes (lower right). (b) The ratio of the plateau modulus
and the corresponding viscous modulus, (G'/G")plateau, grows weakly with increasing crosslinker concentration, and is on average higher for the mutant than for the wild-type at
the same concentration. (c) The frequency at which the elastic plateau occurs (minimum
in G") increases weakly with cross-linker concentration, and is systematically smaller for
the mutant. (d) The relaxation frequency of the network (maximum in G") appears to be
independent of the cross-linker concentration, and is about an order of magnitude smaller
for the mutant.
3.4
Temperature and network dynamics
The networks with mutant a-actinin-4 differ most significantly from their wild-type counterparts in that their relaxation frequencies are consistently lower by at least an order of
magnitude for concentrations ranging from R = 0.0025 to R = 0.04 (Fig. 3-5(d)). We
propose that this shift in time scales is a consequence of a smaller actin/a-actinin-4 unbinding rate for the mutant. Since the equilibrium dissociation constant Kd is the ratio of the
dissociation rate, or off-rate k_, and the association rate, or on-rate k+, we cannot say with
certainty which of the rates differs between wild-type and mutant. However, Weins et al.
[25], following an argument of Wachsstock et al. [21], have explained their observation of
a stronger tendency of wild-type a-actinin-4 to bundle actin partly by assuming a higher
dissociation rate for the wild-type, consistent with the higher value they determined for Kd.
If we adopt this assumption, the dissociation rate of the mutant follows to be about six times
smaller than that of the wild-type. Given that our practical definition of freiax as the frequency at which G" reaches a local maximum only approximates the frequency at which the
relaxation actually occurs, this factor is consistent with the observed factor of slightly more
than 10 between the relaxation frequencies of wild-type and mutant cross-linked networks
(Figs. 3-2, 3-3, and 3-5(d)).
The proposed direct relationship between the characteristic relaxation frequency of the
macroscopic network and the microscopic dissociation rates of its constituents is intriguing.
By controlling the dissociation rates, we can test whether the collective network relaxation
is indeed related to the unbinding of cross-linkers. If we assume that the dissociation rate is
limited by the fraction of cross-linkers with sufficient kinetic energies to overcome the activation energy barrier of the reaction, an obvious way to influence this rate is by means of the
temperature. Based on our proposition, we expect that the characteristic frequencies of the
networks increase with increasing temperature. Our measurements of the dynamic viscoelastic response of cross-linked networks for different temperatures verify this dependence, as
demonstrated in Fig. 3-6 for networks with cross-linker concentrations of R = 0.01 at 15 oC
(blue inverted triangles), 25 'C (black circles), and 35 "C (red triangles). Since the relaxation
100
rm~l
CL
Q) 1
100
CU
M3
10,
0.01
1
0.1
10
f[Hz]
Figure 3-6: Frequency dependence of the elastic modulus G' (solid symbols) and the viscous
modulus G" (open symbols) of actin networks with (a) wild-type and (b) mutant a-actinin-4
(molar ratio of a-actinin-4 to actin R = 0.01) at 37 0 C (red triangles), 25'C (black circles), and 15 'C (blue inverted triangles). Similarly to a high cross-linker concentration, a
lower temperature results in a more pronounced elastic plateau. In addition, a temperature
decrease shifts the network relaxation towards lower frequencies.
frequency
(frelax)
rises faster with temperature than the plateau frequency (fplateau), the elas-
tic plateau spans a wider frequency range at lower temperatures. Moreover, a decrease in
temperature-similarly to an increase in a-actinin-4 concentration-increases the ratio of G'
to G" at the plateau, rendering the network more solid-like. In contrast to a change in crosslinker concentration,which influences the prominence of the elastic plateau while leaving the
relaxation frequency of the network unaltered, a change in temperature appears to influence
both network features. We expect the effect of the mutation to be largely compensated
for by an appropriate temperature increase, such that the viscoelasticity curves of networks
with wild-type and mutant a-actinin-4 at different temperatures coincide without significant
scaling in frequency or moduli. Our observations confirm this notion. For example, we find
that the viscoelasticity of the network with wild-type a-actinin-4 at 25
shown in 3-6(a)
(black circles) agrees well with that of the network with mutant cross-linker at 37 shown in
3-6(b) (red triangles); a direct comparison is provided in 3-7(a). Similarly, the viscoelastic
responses of networks with wild-type (black circles) and mutant (red triangles) a-actinin-4
at R = 0.02 and temperatures of 15 'C and 32.5 C nearly coincide, as illustrated in 3-7(b).
To analyze the temperature dependence of the relaxation frequency more quantitatively,
we measure the frequency dependent viscoelasticity at temperatures between 5 C and 40 'C
for six samples of each cross-linker type with relative cross-linker concentrations ranging
from R = 0.00125 to R = 0.08, and extract
frelax
from these curves. For a few samples,
the relaxation frequency falls outside the range of our measurements (0.01 Hz-10 Hz) at the
lowest temperatures (e.g. blue curve in Fig. 3-6(b)), and the corresponding data are not
included in our compilation. We assume that the temperature dependence of the dissociation
rates of wild-type and mutant a-actinin-4 follows the Arrhenius equation:
k_ = ko - exp -
E
a/(kB NAT)
(3.1)
Herein, Ea denotes the molar activation energy which needs to be overcome for the dissociation reaction to take place, and k0 describes the rate at which unbinding is attempted;
kB
and NA are the Boltzmann and Avogadro constants, respectively. In order to test whether
.
_
(a)
_.
_
_ _. _
______·
..
1. .m
..... .
1.
.
.
. w
. .·
I
W.
.
100
rOM
A
rA
C-
(L
OO0O
10
O
0000
___·
(b)
I·
I
I
*
*
i··· mmii
"'I
m m···i mmmiii·
_•,1,11111111111122
100
.C..
D 10
.1.1
0.01
0.1
f[Hz]
1
10
Figure 3-7: An increase in temperature can return the viscoelasticity of networks with mutant
a-actinin-4 (red triangles) to that of networks with wild-type cross-linker (black circles), as
illustrated by the frequency dependence of G' (solid symbols) and G" (open symbols) for
(a) R = 0.01, wild-type at 25°C, mutant at 37°C, and (b) R = 0.02, wild-type at 15 0 C,
mutant at 32.5 oC. In contrast to the scaling behavior for mutant and wild-type at different
concentrations, a shift in frequency or moduli is not necessary.
the macroscopic relaxation frequencies of the networks exhibit the same temperature dependence, we plot frelax on a natural-logarithmic scale over the inverse of the thermal energy
kB NA
T, measured in Kilojoules per mole. The relaxation frequencies for wild-type (black
circles) and mutant (red triangles) a-actinin-4 form two distinct bands of data points, separated by about an order of magnitude, as shown in Fig. 3-8. The width of these bands
can be attributed to the various cross-linker concentrations in the samples. However, since
the concentration dependence of the relaxation frequency is very weak, we do not distinguish between different concentrations in our plot. The temperature dependent relaxation
frequencies of wild-type and mutant can indeed be described by the Arrhenius equation,
as indicated by straight line fits with slopes of (119.24.4) kJ/mol for the wild-type (black)
and (126.28.0) kJ/mol for the mutant (red). These values are on the order of typical protein interaction energies, allowing us to identify them with the activation energies of the
microscopic dissociation rates.
The disparity in equilibrium dissociation constants Kd between wild-type and mutant aactinin-4 can plausibly be accounted for by a difference in the activation energies. As Weins
et al. have established, the binding surface of the mutant cross-linker is extended by a third
binding site, which is buried in the wild-type. This additional site could simply decrease
the energy level of the bound state, and thus increase the energy barrier Ea for unbinding.
Since the activation energy appears in the exponential, a small difference Ea,mut - Ea,wt of
4.5 kJ/mol suffices to cause a six times lower dissociation rate k-,mut consistent with the lower
equilibrium dissociation constant observed for the mutant [25]. This difference in activation
energies is consistent with our data.
An alternative explanation lies in sequential binding of the two sites that the mutant
a-actinin-4 has in common with the wild-type, and its additional binding site, as described
by the following chemical reaction:
a + Act - (a - Act)
(a - Act)*
(3.2)
We assume that the first reaction, including its association and dissociation rates, is identical
2.72
1.00
0.37
N
I
0.14
.
0.05
0.02
0.01
0.38
0.40
0.42
0.44
1/(RT) [mol/kJ]
Figure 3-8: Temperature dependence of the network relaxation frequency. The logarithm
of the frequency is plotted as a function of the inverse thermal energy. For both wild-type
(black circles) and mutant (red triangles) cross-linker, the relaxation frequency follows the
Arrhenius equation. The slopes, which correspond to an activation energy barrier in the
Arrhenius equation, are comparable. Relaxation frequencies for the mutant are consistently
lower by about an order of magnitude.
for wild-type and mutant. The second reaction only applies to the mutant, and describes its
binding to actin at the third binding site, which results in a distinct, energetically favorable
bound state. The overall concentration of mutant cross-linker, [a]tot, is in this case:
[O]tot
[Oa]free + [a - Act] + [a - Act*] = []fee + [a - Act].
=
=
[afree.-( 1+ k [Act]
1+
(3.3)
(1±)
k*-)
The dissociation constant Kd,mut, as measured in a cosedimentation assay from the amount
of free and bound cross-linker, can be calculated from the dissociation constant for the
wild-type and the reaction rates for the second binding step:
Kd,mut
[a] free [Act]
]free
[a]tot - [a]free
[Act]
[Act]. (i +
.[Act] -
k)
1+
Kd,wt
+
+
(3.4)
In order to recover the experimentally determined factor of six between the binding affinities
of mutant and wild-type, we need to assume a ratio k*l/k* = 5. This ratio also implies that
most of the mutant is bound to actin at all three binding sites, and only one sixth is in the
weaker bound state. We can now explain the lower relaxation frequency of networks with
mutant a-actinin-4 by relating it to the number of cross-linkers that unbind per unit time,
which is the product of k_, the rate at which weakly bound mutant a-actinin-4 dissociates
completely from actin, and the number of mutant in this weaker binding state. With this
interpretation, the shift in relaxation frequencies between wild-type and mutant stems from a
lower prefactor for the mutant, which reflects the smaller fraction of cross-linker available for
unbinding and is only weakly temperature dependent. The main temperature dependence of
the relaxation frequency is determined by the exponential in k_, which is the same for wildtype and mutant. Consequently, the slope of the natural logarithm of frela. over 1/kB NA T,
which is given by the corresponding activation energy Ea, is the same for both types of
cross-linker as well. Our measurements of the temperature dependent macroscopic relaxation
frequencies of networks with wild-type and mutant do not provide a criterion by which to
choose one of the proposed explanations over the other. Both are consistent with our data,
and support the suggested relationship between the network dynamics and the microscopic
binding dynamics.
Chapter 4
Conclusion
We have established through rheological measurements that the dynamic viscoelasticity of
actin networks formed in vitro in the presence of the cross-linking protein a-actinin-4 is
characterized by an elastic plateau in the frequency range between 0.1 Hz and 10 Hz, and a
relaxation towards fluid properties at lower frequencies. These features result from a combination of the properties of individual actin filaments and the cross-linker dynamics. The
high network elasticity at the plateau is a direct consequence of cross-linking, as evidenced
by the increased elastic plateau modulus and greater prominence of the plateau at higher
cross-linker concentration.
In contrasting the effects of the wild-type cross-linker and a mutant known to have
a significantly higher actin binding affinity, we have found the network dynamics for the
mutant to be shifted to longer time scales. In particular, networks with mutant ac-actinin4 undergo their relaxation at lower frequencies. We explain this phenomenon by proposing
that the relaxation of the network on the macroscopic level results from the dissociation of aactinin-4 from actin, which we assume to take place at a lower rate for the mutant, consistent
with its smaller equilibrium dissociation constant. In strong support of this explanation, we
have shown that the temperature dependence of the network relaxation frequency follows
the Arrhenius equation for rate constants of chemical reactions, and allows us to extract a
conceivable value of the energy implied in the dissociation reaction. Based on knowledge
about an additional actin binding site exposed in the mutant cross-linker, we suggest two
different binding mechanisms: one-step binding with a lower energy level in the bound
state for the mutant, and sequential two-step binding, which implies the same temperature
dependence, but different prefactors for the dissociation of wild-type and mutant ac-actinin-4
from actin.
In addition to shifting the relaxation to lower frequencies, the substitution of wild-type by
mutant a-actinin-4 and a decrease in temperature both result in a more pronounced elastic
plateau, thus rendering the network more solid-like. In fact, a higher temperature appears
to compensate for the effect of the mutation.
An increase in cross-linker concentration
likewise alters the frequency dependent viscoelasticity, up to scaling factors in frequency in
modulus, in a manner remarkably similar to the change from wild-type to mutant. Whether
this apparent scalability has a profound reason that could provide further insight into the
properties of cross-linked networks remains a question for future investigations.
Our comparison of actin networks cross-linked with mutant and wild-type a-actinin-4
is not the first study of the effect of different binding affinities of actin cross-linkers on the
dynamic rheological behavior of the resulting networks. Previous comparative studies have
been performed, for example, by Wachsstock et al. [22], who have related the different crosslinker dynamics of Acanthamoeba and chicken smooth muscle a-actinin to the mechanical
properties of the networks. However, the difference between the two cross-linking proteins
studied here is unique in that it results from the smallest possible variation in the underlying
genetic code-a point mutation. Interestingly, we have found this mutation to have a drastic
effect on the properties of in vitro networks at shear frequencies around 1 Hz, close to the
human heart rate. In vivo, the pathological consequences of the mutant cross-linker are
most apparent in podocytes, which are subject to substantial shear stresses resulting from
dynamic capillary pressure. The combination of these facts strongly suggests that altered
viscoelastic properties of cross-linked actin networks contribute to the phenotypic changes
in diseased kidneys.
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